Discussion
Consider the infinite series
\[ (P): \sum_{n=2}^{\infty} \frac{1}{(n \log n)^{1/n}} \]
\[ (Q): \sum_{n=1}^{\infty} \frac{n^{n}}{(2n)!} \]
Then which one of the following statements is correct ?
Answer: D
Series (P) diverges and series (Q) converges.
(P): the terms do not tend to zero. Since \(\ln(n\log n)/n \to 0\), \((n\log n)^{1/n} \to 1\), so each term tends to 1 and the series diverges by the n-th term test.
(Q): apply the ratio test:
\[ \begin{aligned} \frac{a_{n+1}}{a_{n}} &= \frac{(n+1)^{n+1}}{n^{n}}\cdot\frac{1}{(2n+2)(2n+1)} \\ &= \left(1+\frac{1}{n}\right)^{n}\frac{n+1}{(2n+2)(2n+1)} \to e \cdot 0 = 0 \end{aligned} \]
The ratio tends to 0, so (Q) converges: the factorial in the denominator grows much faster than \(n^{n}\). The other options misjudge one of the two series.
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